The period of the pendulum is directly proportional to the square root of the length of the string. The period of such a pendulum with string of length 16 cm is 52 seconds. Find the length of the string if the period is 65 seconds :

Options

A. (A) 4.5 cm
B. (B) 5 cm
C. (C) 6 cm
D. (D) none of these

Answer

To solve the problem step by step, we will use the information given about the pendulum's period and its relationship with the length of the string.

### Step 1: Understand the relationship
The period \( t \) of the pendulum is directly proportional to the square root of the length \( L \) of the string. This can be expressed mathematically as:
\[ t \propto \sqrt{L} \]
This means that:
\[ t = k \sqrt{L} \]
where \( k \) is a constant of proportionality.

### Step 2: Use the given values to find \( k \)
We know that when the length \( L = 16 \) cm, the period \( t = 52 \) seconds. Substituting these values into the equation gives:
\[ 52 = k \sqrt{16} \]
Calculating \( \sqrt{16} \) gives us 4, so we can rewrite the equation as:
\[ 52 = k \cdot 4 \]
Now, solve for \( k \):
\[ k = \frac{52}{4} = 13 \]

### Step 3: Set up the equation for the new period
Now we need to find the length of the string when the period \( t = 65 \) seconds. We can use the same formula:
\[ t = k \sqrt{L} \]
Substituting \( t = 65 \) and \( k = 13 \):
\[ 65 = 13 \sqrt{L} \]

### Step 4: Solve for \( \sqrt{L} \)
To isolate \( \sqrt{L} \), divide both sides by 13:
\[ \sqrt{L} = \frac{65}{13} \]
Calculating the right side gives:
\[ \sqrt{L} = 5 \]

### Step 5: Solve for \( L \)
Now, to find \( L \), we square both sides:
\[ L = 5^2 = 25 \text{ cm} \]

### Final Answer
The length of the string when the period is 65 seconds is:
\[ \boxed{25 \text{ cm}} \]